Preliminary study on parameterization of raw electrical bioimpedance data with 3 frequencies

This study tests the geometrical parameterization method for Electrical Bio-Impedance Spectroscopy (EBIS) readings previously proposed by one of the authors. This method uses the data of just three frequencies (therefore called 3P method). The test was carried out by the analysis of parameterization from 26 spectra (selected from 13 data sets) by the non-linear square (NLS) method, the 3P method and a combination of the two (3P-NLS). Additionally, the behaviour of the 3P method for 4 levels of noise and 3 different ways of segmenting the spectra were also explored with a MATLAB simulation of 400 spectra. Finally, a system for the classification of EBIS readings is presented, based on deviations of the raw data from the semi-circle obtained by the parameterization methods. Overall, the results suggest a very good performance of the 3P method when compared with the other two. The 3P method performs very well with levels of noise of 1 and 2%, but performs poorly with levels of noise of 5% and 10%. The results support the idea that the 3P method could be used with confidence for the parameterization of EBIS spectra, after the selection of three adequate frequencies according to specific applications.


A. Mathematical description
The Cole equation is presented as equation (1), which allows the modelling of the electrical response of a tissue presenting just one dispersion.
Where: is the impedance measured from the biological tissue; is the angular frequency; ∞ is the resistance of the tissue at infinity frequency (resistance of the whole volume, i.e., extra-and intra-cellular spaces); is the resistance of the tissue at zero frequency (resistance of the extracellular space); is the time constant or relaxation time of the tissue, a variable dependant of the capacitive effect of the cell membranes and the resistances of both the intra-and the extra-cellular spaces; is a measure of the relaxation time distribution, which can be related to different phenomena as, for instance, cellular or molecular interactions, cell size, anisotropy or heterogeneity of the tissue, among others [1]- [3]. This equation corresponds to a semi circumference in the complex plane, and, therefore, a relation between its parameters and those of the semi circumference can be obtained, as it has been reported by [4] and [5].
The three parameters of the circle are named, here, as h (abscise), k (coordinate) and r (radius).
The method used for the parameter estimation is as follows: Three points of the impedance spectra are taken in the complex plane and the circle equation calculated.
R 0 and R ∞ are calculated using the parameters of the circumference (centre and radius, i.e., h, k and r), while α is calculated using the parameters h and r).
τ is calculated replacing R 0 , R ∞ and α in (1) and using the selected impedance values and their respective frequencies.
For the calculation of the equation of the circle passing through the three selected points, the starting point is given for the general equation of the circle (equation 2), with three unknown values (h, k and r). As there are three known points [(x 1 , y 1 ), (x 2 , y 2 ) and (x 3 , y 3 )], a three equation system with three unknowns can be formulated: Equation (2) can be rewritten as: = −2ℎ; = −2 = ℎ + − (4).
The parameter equals to: where ∅ is the normalized angle formed by the abscise and the radius touching R ∞ . In other words: = 1 ± atan √ (9).
Parameter is not geometry dependant and, therefore, the impedance values are used in order to find it. Rearranging equation (1), it follows that: Here, replacing Z, its corresponding frequency (ω) and the previously obtained Cole parameters, the value for τ is calculated. Due to possible measurement errors and the trunction process in the handling of the data, the result is complex and therefore, only the real part is taken into account. Equation (10) can be easily evaluated by means of either a computer or a processor. Nevertheless, in order to facilitate the calculation without the use of complex numbers, following expression applies to the real part: Where and correspond to the real and the imaginary parts of the impedance measured, respectively, as indicated in equation (12).

= + (12).
Given that there are three points or impedance measurements, a value for τ for each one is found and then the three values are averaged.
Equations (10) and (11) were obtained from complex algebraic deductions from the Cole equation.

Subset Tail Head Subset
Tail Head Subset Tail Head                        In the Tables S20 to Table S25, NL: Noise Level, SP: Scatter Plot linear parameters, m is the slope, b is the y-intercept and r is de Pearson correlation coefficient of the linear fit for the data, xi: abscise i, yi: ordinate i.
The higher values of r are shown in bold.